# Defining Return

It is important to define the distinction in objectives when considering the performance of an investment. The first distinction is between performance measurement and performance evaluation. Measurement is simply an accounting function, which attempts to reconcile end-of-period with initial values. Evaluation addresses the more difficult issues of how the measured return was achieved; whether it was due to skill and/or luck, and whether future outcomes will be similar. There are many ways of measuring returns. The method chosen depends upon the particular performance evaluation objectives. If the performance of an investor's portfolio is being evaluated, for example, then the dollar-weighted rate-of-return would be appropriate, as it provides a measure of return from the perspective of the portfolio's owner. Alternatively, if a fund manager's decision-making is being evaluated and, to avoid obfuscating the results of his judgement the effect of the investor's cash flow decisions is included, then the time-weighted rate-of-return methodology should be used.

The dollar-weighted rate-of-return is the internal rate-of-return that equates the initial contribution of investments and the cash flows that occur, with the end value of the investment. Mathematically, this measure of return is the dollar-weighted average of sub-period returns, with the dollar weights equal to the sum of the initial contribution and all of the cash flows up to the time of the sub-period return, multiplied by the length of that sub-period. Its strengths are twofold. It is easy to understand and compute, and provides the true rate-of-return from an investor's perspective. A weakness is that it may cloud the effect of a fund manager's decisions, because the intervening cash flows strongly influence the calculation of this return, and these cash flows are typically independent of the manager's decisions.

The time-weighted rate-of-return is the weighted average of the internal rates-of-return for the sub-periods between the cash flows, weighted by the length of those sub-periods. One problem with the time-weighted rate-of-return is that, to be calculated precisely, not only the time and amount of each cash flow is required, but also the value of the investment at the time of each cash flow. That is problematic for many reasons, primarily as some assets are difficult to value, real estate being a good example. The same calculation procedures are often used for all asset classes, with the typical assumption that all cash flows occur at the end of each month; this results in returns that are not based on valuations at the precise dates of the cash flows. Therefore, methods of approximation are used to calculate these time-weighted rates-of-return, thus introducing errors into the calculation. This occurs particularly in real estate investment; Giliberto [1988; 1994] considers this point.

There is bias in all calculation methods, but some are preferred to others. The theoretical strength of the methodology of the time-weighted rate-of-return is that it gives the true measure of returns due solely to the manager's decisions. It thus eliminates the arbitrary effect of the investor's cash flow decisions. It seems clear that the time-weighted rate-of-return is much better than the dollar-weighted rate-of-return for evaluating manager performance.

Alternatively, the investor may want to understand the effects of strategic allocation decisions. Here, some hybrid between time-weighted and value-weighted rates-of-return may be more appropriate, so that only the cash flows that are part of the strategic decisions are appropriately accounted for, those others being ignored so as not to influence the measured rate-of-return. When considering returns over multiple time periods it is usual to use a method of averaging the returns for the individual time periods. The two common constructs are the simple arithmetic average return and the geometric compound rate-of-return. The geometric compound rate-of-return is the rate that equates the value at the beginning of the first period with the value at the end of the last period, so it is the rate-of-return experienced over the full period. While the arithmetic average return does not provide the rate-of-return of a period, it has other useful properties; see section 4.3.1.

Performance figures are often used to form expectations for future performance, either for investment strategy, or for the evaluation of a fund manager. For these applications, the arithmetic average return is superior to the geometric compound return. Indeed, the arithmetic average return is often used as an unbiased forecast of the next period's return. It is also the preferred method for selecting an investment or manager when the performance during the next period is under consideration. However, if it is long-term performance expectations which are of interest, use should be made of the geometric compound return. For example, consider an investment that is up 50% in the first year and down 50% in the second year. The arithmetic average return is zero percent, but the geometric compound return is -13.4%, reflecting the true return from this performance. If this continues to be up 50%, then down 50%, year after year, the arithmetic average return will continue to be zero percent, and this will always be the appropriate forecast of the next period's return. Over a long period of time, however, the geometric compound return tends towards -100%. This emphasises the importance of understanding the return methodology used when evaluating an investment.

## Costs

Costs, in particular those transaction related, are another element of performance. The impact on investment value of costs is often ignored when measuring the current value of an investment. Deductions for e.g. transactions costs, in order to return the investment to its original (cash) basis, are often ignored. When valuing equities, it is usual to find closing prices being used without adjustment for the transaction costs that must be incurred in order to return cash to investors. With some investments, illiquidity or excessive turnover generate high transactions costs. Direct real estate investment is a prime example of this, with transaction costs of five percent or more; see section 2.4.4. Therefore, the assumption that a sale could have been made at the observed historical valuation may be extremely misleading.